Han's conjecture on Hecke compatibility of geometric Eisenstein series

Let $P$ be a parabolic subgroup of $G$ with Levi subgroup $M$ and unipotent radical $U$, let $V\in\operatorname{Rep}({}^LG)$, and restrict $V$ to ${}^LP$. Choose a finite filtration

$$0=V_0\subset V_1\subset\cdots\subset V_m=V|_{{}^LP}$$

such that the ${}^LU$-action on each graded piece $W_i=V_i/V_{i-1}$ is trivial, so that each $W_i$ is naturally inflated from $\operatorname{Rep}({}^LM)$. Han's conjecture. The functor $T_V\operatorname{Eis}_P(-)$ admits a corresponding finite filtration whose graded pieces are $\operatorname{Eis}_P(T_{W_i}-)$. This predicts compatibility between geometric Eisenstein series and Hecke operators, and is used in the source to study the case $G=\operatorname{GL}_n$ and extended pure inner forms. Its status is not resolved in the supplied text.